Table 2 Table of discrepancies $M^{⊺}{U}_{(i,j,k)}=M^{⊺}(X_{(i,j,k)}-X)$ for the 3-state Markov process. The discrepancy $M^{⊺}{U}_{(1,2)}$ (marked by ∗) corresponds to reduced process $X_{(1,2)}$ projected onto the third component, which is the optimal two-edge-neglecting approximation of X for this example, in agreement with Schmandt and Galán [14]

$M^{⊺}{U}_{(1)}$

$R_1$

0.0417

$M^{⊺}{U}_{(2)}$

$R_2$

0.0417

$M^{⊺}{U}_{(3)}$

$R_3$

0.2917

$M^{⊺}{U}_{(4)}$

$R_4$

0.2917

$M^{⊺}{U}_{(1,2)}$

$R_1+R_2$

0.0833*

$M^{⊺}{U}_{(3,4)}$

$R_3+R_4$

0.583

$M^{⊺}{U}_{(1,3)}$

$R_1+R_3$

0.3333

$M^{⊺}{U}_{(1,4)}$

$R_1+R_4$

0.3333

$M^{⊺}{U}_{(2,3)}$

$R_3+R_3$

0.3333

$M^{⊺}{U}_{(2,4)}$

$R_2+R_4$

0.3333

$M^{⊺}{U}_{(1,2,3)}$

$R_1+R_2+R_3$

0.375

$M^{⊺}{U}_{(1,2,4)}$

$R_1+R_2+R_4$

0.375

$M^{⊺}{U}_{(1,3,4)}$

$R_1+R_3+R_4$

0.625

$M^{⊺}{U}_{(2,3,4)}$

$R_2+R_3+R_4$

0.625